Global Journal of Pure and Applied Mahemaics. ISSN 0973-1768 Volume 13, Number 9 (017, pp. 6695 670 Research India Publicaions hp://www.ripublicaion.com/gjpam.hm Pricing formula for power uano opions wih each ype of payoffs a mauriy Youngrok Lee Deparmen of Mahemaics, Sogang Universiy, Seoul 04107, Korea. Hyeon-Seung Yoo Deparmen of Mahemaics, Sogang Universiy, Seoul 04107, Korea. Jaesung Lee Deparmen of Mahemaics, Sogang Universiy, Seoul 04107, Korea. Absrac Nowadays, power opions become widely used in derivaive markes on foreign exchange, which means he combinaion of uano opion and power opion can be seriously considered on is opimal pricing. Here, we drive full closed-form expressions for he price of a European power uano call opions wih four differen forms of erminal payoff under he assumpion of he classical log-normal asse price and exchange rae model. AMS subjec classificaion: Primary 91B5, 91G60, 65C0. Keywords: power opion, uano opion, payoffs a mauriy, explici formula.
6696 Youngrok Lee, Hyeon-Seung Yoo, and Jaesung Lee 1. Inroducion Power opions are a class of exoic opions in which he payoff a mauriy is relaed o he cerain posiive power of he underlying asse price, which gives he buyer of he opion he poenial o receiving a much higher payoff han ha from a vanilla opion. Since a small change in he value of he underlying of a power opion may lead o a significan change in he opion price, power opions are widely used in markes o provide high leverage sraegy. A uano is a ype of financial derivaive whose pay-ou currency differs from he naural denominaion of is underlying financial variable. A uano opion is a crosscurrency opion which has a payoff defined wih respec o an underlying in one counry, bu he payoff is convered o anoher currency for paymen. For example, an invesor rades a uano opion on an asse uoed in foreign currency, a mauriy, he opionâł s value in foreign currency will be convered a a fixed rae ino domesic currency. Indeed, uano opions may have four differen forms of erminal payoffs. To menion each ype of payoffs a mauriy, i is a foreign euiy opion convered o domesic currency, a foreign euiy opion sruck in domesic currency, a foreign euiy opion sruck in pre-deermined domesic currency or an FX opion denoed in domesic currency, respecively. There have been many researches on pricing power opions or uano opions, boh of which are relaively easy under he classical Black-Scholes [1] model. To menion some references on hose subjecs, Heynen and Ka [3] and Tompkins [7] focused on power opions boh heoreically and from he marke poin of view. Wysup [9] obained various uano opion price formula and inroduced hree vega posiions on hedging of uano opions, and exbooks of Wysup [8] and Kwok [4] conain general heory of uano opions. Since power opions become widely used in derivaive markes on foreign exchange, he combinaion of uano opion and power opion can be seriously considered on is pricing. Alhough we can no find any previous research on power uano opion pricing, here is a research paper on pricing power exchange opion by Blenman and Clark [], on which he auhors combine power opion and Margrabe [6] ype exchange opion. In his paper, we drive full closed-form expressions for he price of power uano call opions wih four differen forms of erminal payoff under he assumpion of he classical log-normal asse price and exchange rae model. In secion, we specify he dynamics of he processes of underlying asse price and exchange rae in he risk-neural world. In secion 3, we specify four differen ypes of payoff a mauriy and obain he analyic expressions for he price of a power uano call opion in each case.. Risk-neural dynamics on currencies For a dividend paying asse wih he dividend yield rae, le S be he asse price in foreign currency X, and le V be he foreign exchange rae in foreign currency per uni of he domesic currency wih he consan volailiies σ S and σ V, respecively. which
Pricing formula for power uano opions wih each ype of payoffs a mauriy 6697 have he following risk-neural dynamics: ds = ( r f S d + σ S S db Qf, dv = ( r f r d V d + σ V V dw Qf (.1 under he risk-neural probabiliy measure Q f, B Qf and W Qf are wo sandard Brownian moions in foreign currency wih he correlaion ρ. Also, r f and r d are he consan foreign and domesic riskless raes, respecively. Then from well-known sandard procedure (see page 95 of [8] or secion of [5], we ge he risk-neural dynamics of (.1 in domesic currency are given by B Qd and W Qd ds = ( r f ρσ S σ V S d + σ S S db Qd, dv = ( r d r f V d + σ V V dw Qd, (. are wo correlaed sandard Brownian moions in domesic currency. 3. Quano opion prices for various uano payoffs The following heorems in nex subsecions give he explici formulas for he prices of European power uano call opions wih consan foreign and domesic riskless raes according o four differen forms of erminal payoffs menioned in secion 1. 3.1. Type I The foreign euiy power- uano call opion (sruck in foreign currency convered o domesic currency has a mauriy payoff given by K f is he foreign currency srike price. V T max ( S T K f, 0, (3.1 Theorem 3.1. Under he assumpions of (. wih > 0, he price of a European power- uano call opion a ime in domesic currency wih he payoff (3.1 is given by C (1 (,V = V e r f (T S e r f + ( 1σ S (T N (d1 K f N (d, ln S K f + r f + ( 1σ S (T σ S T, d = d 1 σ S T
6698 Youngrok Lee, Hyeon-Seung Yoo, and Jaesung Lee and N ( denoes he cumulaive disribuion funcion for he sandard normal disribuion. Proof. We may wrie (1 as (1 (,V = e r d (T [ E Q d VT max ( ST K f, 0 ] F = V e r f (T E Q d ( e σ V (T +σ V W Qd T W Qd max ( ST K f, 0 F. (3. For a new risk-neural probabiliy measure Q d, he Radon-Nykodým derivaive of Q d wih respec o Q d is defined by d Q d dq d = e σ V +σ V W Qd. F Then he Girsanov s heorem implies ha B Q d = B Qd ρσ V is again a sandard Brownian moion under he domesic risk-neural probabiliy measure Q d. Moreover, he dynamics of S under he measure Q d is given by ds = r f + ( 1 σ S S d + σ SS Thus, (3. becomes (,V = V e r f (T [ ( E Q max S d T K f, 0 ] F C (1 = V e r f (T S r f + ( 1σ S e db Q d. (3.3 (T N (d1 K f N (d, ln S K f d = d 1 σ S T. + r f + ( 1σ S (T, σ S T 3.. Type II The foreign euiy power- uano call opion sruck in domesic currency has a mauriy payoff given by max ( V T S T K d, 0, (3.4
Pricing formula for power uano opions wih each ype of payoffs a mauriy 6699 K d is he domesic currency srike price. Theorem 3.. Under he assumpions of (. wih > 0, he price of a European power- uano call opion a ime in domesic currency wih he payoff (3.4 is given by ( ( [ r,v = V S e f + r f + ( 1σ S ] (T N (d1 K d e r d(t N (d, [ ] ln V S K d + r d r f + r f + ρσ S σ V + ( 1σ S + σ V (T ( σs + σ V + ρσ, Sσ V (T d = d 1 ( σ S + σ V + ρσ Sσ V (T. Proof. We may wrie C ( as ( ( [ ],V = e r d (T E Q d max (ŜT K d, 0 F, (3.5 Ŝ T = V T S T. We noe ha he risk-neural dynamic of Ŝ in domesic currency is given by dŝ = [ r d r f + r f + ( 1 σ S from (. and (3.3. Thus, (3.5 becomes ] Ŝ d + σ S Ŝ db Qd + σ V Ŝ dw Qd ( ( [ r,v = V S e f + r f + ( 1σ S ] (T N (d1 K d e r d(t N (d, [ ] ln V S K d + r d r f + r f + ρσ S σ V + ( 1σ S + σ V (T ( σs + σ V + ρσ, Sσ V (T d = d 1 ( σ S + σ V + ρσ Sσ V (T.
6700 Youngrok Lee, Hyeon-Seung Yoo, and Jaesung Lee 3.3. Type III A a foreign euiy power- uano call opion sruck in pre-deermined domesic currency has a mauriy payoff given by V 0 max ( S T K f, 0, (3.6 V 0 is he some fixed exchange rae and K f is he foreign currency srike price. Theorem 3.3. Under he assumpions of (. wih > 0, he price of a European power- uano call opion a ime in domesic currency wih he payoff (3.6 is given by C (3 ( = V0 e r d(t S e r f ρσ S σ V + ( 1σ S (T N (d1 K f N (d, ln S K f d = d 1 σ S T. + r f ρσ S σ V + ( 1σ S (T, σ S T Proof. We may wrie (3 as (3 ( = V0 e r d(t [ ( E Q d max S T K f, 0 ] F = V 0 e r d(t E Q d S e r f ρσ S σ V + ( 1σ S (T N (d1 K f N (d, ln S K f d = d 1 σ S T. + r f ρσ S σ V + ( 1σ S (T, σ S T 3.4. Type IV An FX power- call opion denoed in domesic currency is an euiy-linked foreign exchange opion which has a mauriy payoff given by S T max (V T K e, 0, (3.7
Pricing formula for power uano opions wih each ype of payoffs a mauriy 6701 K e is he srike price on he exchange rae. Theorem 3.4. Under he assumpions of (. wih > 0, he price of a European power- uano call opion a ime in domesic currency wih he payoff (3.7 is given by (4 (,V = S e [ r f ρσ S σ V + ( 1σ S r d ] (T V e (r d r f +ρσ S σ V (T N (d 1 K e N (d, ln V K e + (r d r f + ρσ S σ V + σ V (T, σ V T d = d 1 σ V T. Proof. We may wrie C (4 as (4 (,V = e r d (T [ E Q d S T max (V T K e, 0 ] F [ = S e r f ρσ S σ V + ( 1σ S ] r d (T ( E Q d e σ S (T +σ S B Qd T BQd max (V T K e, 0 F. (3.8 For a new risk-neural probabiliy measure Q d, he Radon-Nikoým derivaive Q d wih respec o Q d is defined by d Q d dq d = e F σ S +σ S B Qd. Then he Girsanov s heorem implies ha B Q d = B Qd σ S and W Q d = W Qd ρσ S are again sandard Brownian moions under he domesic risk-neural probabiliy measure Q d. Moreover, he dynamics of V under he measure Q d is given by dv = ( r d r f + ρσ S σ V V d + σ V V d W Q d.
670 Youngrok Lee, Hyeon-Seung Yoo, and Jaesung Lee Thus, (3.8 becomes (,V = e r d (T [ E Q d S T max (V T K e, 0 ] F C (4 [ = S e r f ρσ S σ V + ( 1σ S ] r d (T E Q d [max (V T K e, 0 F ] = S e [ r f ρσ S σ V + ( 1σ S r d ] (T V e (r d r f +ρσ S σ V (T N (d 1 K e N (d, ln V K e + (r d r f + ρσ S σ V + σ V (T, σ V T d = d 1 σ V T. References [1] F. Black and M. Scholes, The pricing of opions and corporae liabiliies, The Journal of Poliical Economy, 81 (1973, no. 3, 637 654. [] P. Blenman and P. Clark, Power exchange opions, Finance Research Leers, (005, no., 97 106. [3] R. Heynen and H. Ka, Pricing and hedging power opions, Financial Engineering and he Japanese Markes, 3 (1996, 53 16. [4] Y.-K. Kwok, Mahemaical models of financial derivaives, nd ed., Springer, Berlin, 008. [5] Y. Lee and J. Lee, Local volailiy for uano opion prices wih sochasic ineres raes, Korean J. Mah. 3 (015, no. 1, 81 91. [6] W. Margrabe, The value of an opion o exchange one asse for anoher, Journal of Finance. 33 (1978, no. 1, 177 186. [7] R. G. Tompkins, Power opions: hedging nonlinear risks, (1999, no., 9 45. [8] U. Wysup, FX Opions and Srucured Producs, The Wiley Finance Series, 007. [9] U. Wysup, Quano Opions, MahFinance AG (008, 1 1.